On infinite regions

Abstract

The object of the present paper is to lay down a general definition of infinite regions which will include the cases of projective geometry, the geometry of inversion, the geomnetry of the space of analysis, and other geometries which, like these, mnay be based on a set of elements consisting of the points of ordinary space of n complex dimensions, extended by a complex (n -1 )-dimensional set of ideal elements-the so-called points at This latter manifold is algebraic in character, being in the familiar cases the line or plane at infinity, the null circle or sphere at infinity, or the n hyperplanes at infinity. We are not concerned directly with geometries based on ordinary real space-real projective geometry, the real geometry of inversion, etc. In these geomnetries the infinite region is sometimes a mnanifold of only one less dimension than the space it closes, and the extended space is then not necessarily linearly simply connected. Here, however, the space considered is of 2n real dimensions, and it is closed by a manifold of two less dimensions, the resulting extended space being always linearly simply connected. A prototype for one class of cases included here is given by the geomnetry of inversion, to which reference has already been made. In that geometry a one-to-one relation is established between the elements of the group of circular transformations, whose transformations operate on the points of complex n-dimensional, space, and the group of collineations in space of one higher dimension, which carry a certain quadric surface (or hypersurface, when n > 2) over into itself. Furthermore, a one-to-one relation between the points of the n-dimensional space and the points of the quadric is established. Now the quadric is taken in projective space, and it is closed by the ideal points in which it meets the plane at infinity. But ordinary n-dimensional space is not closed, and it is not until it has been closed by a suitable set of ideal elements-the so-called infinite region-that such a correspondence is possible. Aside, however, altogether from a correspondence of this kind, it is necessary to introduce the same set of ideal elements if the transformations of the first named group are to carry each point of the set on which they operate into a point of that set.